Aust. J. Phys., 1991, 44, 353-62

Models for Supersymmetric Quantum Mechanics

M. Baake,A R. Delbourgo and P. D. Jarvis

Department of Physics, University of Tasmania, G.P.O. Box 252C, Hobart, Tas. 7005, Australia. A Institut fur Theoretische Physik, Universitat Tubingen, 7400 Tubingen I, FRG.

Abstract An algebraic view of supersymmetric quantum mechanics is taken, emphasising the couplings between bosonic and fermionic modes in the supercharges. A class of model Hamiltonians is introduced wherein the fermionic (bosonic) operators are canonical and the bosonic (fermionic) ones satisfy a Lie algebra (superalgebra) whose representation theory permits the complete solution of the model in principle. The kinematical symmetry of such models is also described. The examples of one and two bosonic models, with 5U(2) and 5U(3) dynamical algebras respectively, are analysed in detail.

1. Introduction and Main Results Supersymmetry provides a unified framework for the exact solutions of classical textbook nonrelativistic quantum mechanical models (Haymaker and Rau 1986; Fuchs 1986). In the simplest version the odd and even parts of the Hamiltonian correspond to 'partner' potentials for the system, whose form is related by supersymmetry and which produce strongly constrained spectra, phase shifts, etc.; in soluble cases the partners are typically reparametrisations of each other. More generally, one can consider conditions under which a wider class of Hamiltonians with a given number of bosonic and fermionic canonical coordinates and momenta possesses simple or extended supersymmetry (de Crombrugghe and Rittenberg 1983; Rittenberg and Yankielowicz 1985), allowing also for constraints (Flume 1985). Making reasonable assumptions about the couplings in the Hamiltonian and supercharges, there are only a limited number of solutions (up to N = 8); the more extended models often require exotic values of magnetic moments, monopole charges and so on, conspiring with highly specific spatial potentials (D'Hoker and Vinet 1985). In the present paper we point out a general class of N = 1 supersymmetric Hamiltonians H = {Q, Qt} which arise from supercharges involving couplings between fermionic and bosonic modes; one set of modes satisfies canonical commutation (or anticommutation) relations while the other set is assumed to satisfy certain Lie algebra (or superalgebra) relations. Primary amongst these is the requirement that they should mutually commute (or anticommute) in order to secure Q2 = Qt2 = O. Further conditions will dictate the type of kinematical and dynamical symmetry possessed by the model Hamiltonian.

0004-9506/91/040353$05.00 354 M. Baake et al.

In the next section we treat the bosonic compact case of one canonical mode (n = 1), which produces an 5U(2) dynamical algebra. * When n = 2 the bosonic compact case leads to an SU(3) dynamical algebra; this is considered in detail in the following section as an example of this class of model. Concluding remarks and generalisations are drawn in Section 4.

2. An SU(2) Example As a simple illustration of the construction, consider the n = 1 case consisting of one bosonic and one fermionic operator:

Q=Ea, Qt =Fat, where

{a,at }= 1, {a,a}={at,at}=O,

and F is the hermitian conjugate of E. The requirement Q2 = Qt2 = ° is now trivially satisfied and H is fixed as

H = EF - [E,F]NF,

where NF = a t a is the number operator for fermionic modes. If we now consider E and F as generators of a Lie algebra, then possible supersymmetric H are enumerated by giving the representation of the Lie algebra and the matrix elements of the various operators therein. The simplest possibility is that E and F generate the angular momentum SU(2):

[E,F] = 21o, Uo,E] =E, Uo,F] =-F.

Introducing now the orthonormal basis states U m} with m = -j,-j+ 1, ... ,j-1,j; j = 0, 1/2, 1, ... and the well-known matrix elements

1-U m}= ~U+m)U-m + 1) U m -l},

we obtain the spectrum of bosonic (nF = 0) and fermionic (nF = 1) sectors,

E5~ = U + m)U - m + 1), E5~ = U - m)U + m + 1).

In each sector this corresponds to a singlet E = ° state (m = -j and m = +j, respectively) and a tower of doubly degenerate E > ° states ranging up to Emax=jU+1) for m=O,l and m=O,-l, respectively.

3. Case Study of n = 2 Turning now to the n = 2 bosonic case we consider the Lie algebra generated by two commuting operators El,E2 and their hermitian conjugates Fl,F2. For example if [Fl,E2] oc El, then the Lie algebra is SU(3) or a non-compact real form thereof. In principle any rank 2 semi-simple Lie algebra is admissible, which then becomes the dynamical or spectrum-generating symmetry for the

• The non-compact 5U(1,I) case is mentioned in the Conclusions. Supersymmetric Quantum Mechanics 355

supercharges

Q=E1al +E2a2' Qt =Fla1 +F2a2 and their associated supersymmetric Hamiltonian; al,a2 and their conjugates aI, a 2 stand for fermionic creation and annihilation operators. In the general bosonic case we would have t _, t Q= ,LEocaoc, Q - LE_()(a()(, oc ()( where the E()( are a set of mutually commuting positive root vectors of the dynamical algebra Land a()(,at are the corresponding set of canonical fermionic operators. The representations of L together with the fermionic Fock space provide the physical Hilbert space of the model; H is in the enveloping algebra of L In addition, there normally is an identifiable kinematical symmetry Xc: L which commutes with H and accounts for some of the degeneracies of the spectrum, apart from the usual supersymmetric doubling of states (Haymaker and Rau 1986). For models generalising Q, Q t above, L is SU(n) and X is SU(n -1); other choices of H lead to different X's. It is our contention that, in contrast to the highly constrained nature of the supersymmetric systems alluded to above, the present type of model may well be realised in simple physical situations, given the simple ansatz taken and the minimal structural assumptions about the bosonic sector. There is thus the hope that models of this type may have a utility akin, say, to the use of group theory for two-level systems, namely SU(2) realised via the Pauli matrices (Dirac 1936). To this extent, the proposal is intermediate between the highly constrained models and models based on explicit supergroup chains, as applied to nuclear spectra (for a review of other applications see Kostelecky and Campbell 1985; Balantekin 1985; Sukumar 1985). We now turn to a detailed examination of the SU(3) case, for both matrix and oscillator realisations.

Fixed Irreducible Representation of SU(3) For ease of notation we introduce the entire set of SU(3) generators in matrix form Ej, 1 ~ i,j ~ 3,

i k k i i k [Ej,E.el = 8j E.e - 8.e Ej, (1)

where I.r=l Ei = 0 and E{ = (Ej)t in unitary representations. We may choose El and E2 as any suitable pair of mutually commuting generators; for example,

El == Et E2 == Et and put a == aI, b == a2 so

Q== aEl +bE2 and H=={Q,Qt}. (2)

Upon using the SU(3) algebra (1), we get

H = EIE~ + EfE~ + (1- Na)(E~ - E~) + (1- Nb)(E~ - E~) + a t bE~ + b t aE~, (3) 356 M. Baake et al.

where Na and Nb are number operators for fermionic modes of type a and b, respectively. In order to describe states in representations of 5U(3) so as to choose a symmetry-adapted basis. for H, we adopt the Gel'fand labelling scheme (Biedenharn and Louck 1981) p p' 0) q q' r where the rows are ordered by lexicality and successive rows are subject to 'betweenness' conditions,

p ~ p' ~ 0, q ~ q' ~ 0, r ~ 0,

p ~ q ~ p', p' ~ q' ~ 0, q ~ r ~ q',

for the integer labels p,p',q,q' and r; the pair {p,p'} labels the highest weight of the representation corresponding to the partition of two parts of length p and p'. A generic state then has weight given by

pp'O pp'O E} I q q' )== [r-(p+p')/3] q q' ), r r

p p' 0) p p' 0) E~I q q' ==[q+q'-r-(p+p')/3] q q' , (4) r r

PP'O) PP'O) E~ I q q' == [2(p + p')/3 - q - q'] q q' . r r

Matrix elements of other E) are then prescribed; for instance (Biedenharn and Louck 1981)

p p' p p' ~ ( + 1 ° ' E1 , 0) == (p - q)(q - p' + 1)(q + 2)(r - q, + 1) . (5) q q 3 q q (q _ q' + 1)(q - q' + 2) r+ 1 r

From (2) and (3) the Hamiltonian H acts on the tensor product :F ® V{P,P'} of the 4 dimensional fermionic Fock space :J with the bosonic carrier space Y of the 5U(3) representation with highest weight {p,P'}. Diagonalisation can be carried out explicitly using (5) and similar matrix elements; however the analysis is simplified by identifying which operators provide the kinematical or degeneracy symmetry of the problem by commuting with H. From the form of H it is clear that E~ and NF == Na+Nb are two such operators. Moreover, Q itself can be regarded as a scalar under transformations of the 2,3 indices if there are also contragedient transformations of a,b. To this end, Supersymmetric Quantum Mechanics 357

we introduce the V-spin generators (Lichtenberg 1978)

V+=E~, V_=E~, V3 = (E~ - E~)/2 (6) and their fermionic counterparts'

v+ =-bt a , v_=-atb, V3 = (bt b-at a)/2, (7) form the total V-spin operators

11±=V±+v±, 113 =V3+ V3 (8) and confirm that [Y,Q] = [Y,Qt] = 0, (9) where 11± == 111 ±i112 as usual. In order to find the energy spectrum we note that El is diagonal in the Gel'fand-Tseytlin basis, and that states with nF = 0 or nF = 2 have v = 0 and hence 11 = V, whereas states with nF = 1 have v = ~ and need to be combined with states having V = 11 ± ~. However these v = ~ states are accessible from the nF = 0,2 states, with which they are degenerate since [Q,H] = 0 = [Qt,H], by the action of Q or Q t; therefore it is sufficient to examine the spectrum of H in the former cases. For definiteness consider the case p' == O. Then the representation {p,O} contains V-spin multiplets 0,!,I, ... ,p/2 and the states

POO) q 0 r

are diagonal in V and V3 with eigenvalues V=(p-r)/2, V3 = q-(p+ r)/2. (10)

In this case Et and V give equivalent information since from (4) and (9), Et = 2p/3 - 2V. Thus we may label the states as

I nF; na, nb) ® Ip; V, V3} with the understanding that v-spin is given by (7) and is to be coupled with V to total 11 for the energy eigenstates. The resulting character of energy levels is shown in Table 1. States of pure V-spin with v = 0 are designated 10 or 112 depending on nF (which is

Table 1. Spectrum of energy levels in the SU(3) example N=O N=l E

10 (Qt/h 2V(p-2V+l) (QlIh 112 (2V + 2)(P - 2V)

• This assignment makes a,b transform contragrediently to E!,E~ under the total V-spin generators, so that the supercharges become scalars. 358 M. Baake et al.

given as a subscript); these are degenerate with mixed v =~, V = l' ± ~ states designated (Qt 1)1 and (QIl)2. The last column contains the energy eigenvalue; the corresponding eigenvectors are

QtIO,O) ®IV, V3) = ~(V + V3 + l)(p- 2V) 11,0} ®IV +~, V3 +~)

~(V-V3+1)(P-2V) 10,1}®IV+~,V3-~} (11) for type I, and

Qll, 1}®IV, V3}= ~(V + V3)(P- 2V + 1) 10, 1}®1V -~, V3 -~}

+~(V-V3)(P-2V+1) 11,0}®IV-~,V3+~} (12) for type II. Finally, the columns labelled :N = O,:N = 1 refer to the eigenvalue of the operator :N = [Q t, Ql/H which plays the role of the fermion number for states in the positive energy spectrum (Stedman 1985) in applications where spin is not explicitly defined or where the spin-statistics connection is not physically relevant. * The final category of states lfJ has E = 0 and indefinite :N and includes states annihilated by both Q and Qt. From the form (2) of the supercharges, these include both the highest weight state of the 5U(3) representation with nF = 2, V = V3 = 0, POO) 12; 1, I} ® I P 0 P

and the lowest weight state of the SU(3) representation with nF = 0, V = V3 = p/2,

p 00) 10; 0, O} ® I 000

together with the V-spin multiplet in which it lies.§ Finally there are the mixed states of l' = (p+ 1)/2 which obviously cannot be degenerate with pure V = (p+ 1)/2 states; for example, it is readily checked from (5) that the maximal 1'3 state POO) 10, 1}® I P 0 o

is indeed annihilated by both Q and Qt.

* Note that in this basis 'fermion number' N = 0,1 is not directly related to the evenness or oddness of NF. § The V-spin shifting nature of Q and Qt in (10) and (11) is due to the commutators [Et,Q] = Q and [Et,Qt] = _Qt, plus the fact that for the present choice of representation pi == 0, E1 and V are equivalent. Supersymmetric Quantum Mechanics 359

Even states N = 0 Odd states N = 1 Level (E) V El Level (E) V El

= (9) 1/2 5/3

= (8) 0,1 8/3,2/3 C::::::J

~::::::J = (6) 1,3/22/3,-1/3

== (5) 3/2 -1/3

" __ ~ ___ J = (4) 1/2,25/3,-4/3

Zero states Level (E) V El

~ (0) 2,0,5/2 -4/3,8/3,-7/3

Fig. 1. Energy spectrum in the n = 2 case. For assignment of types Io,IIz,(Qtlh and (QIIh states, as well as

This ends the analysis of the model for a fixed irreducible representation of the bosonic Lie algebra. The spectrum derived from Table 1 is given in Fig. 1 for the p = 4 case (li-dimensional representation) for illustrative purposes.

Oscillator Realisations We now turn to an investigation of the model for oscillator realisations in the n = 2 case. While realised in (possibly infinite-dimensional) Fock spaces, these can nevertheless be analysed in terms of the individual representations 360 M. Baake et al.

of the underlying dynamical algebras into which they decompose, as done previously. However it is of more interest to consider the form of the model Hamiltonian from the oscillator viewpoint as it is likely to be useful for applications in such a second-quantised formulation and can even be related to canonical phase space coordinates; moreover there exists a body of literature (Rittenberg and Yankielowicz 1985) on supersymmetric quantum mechanics with which the present ansatz may be compared. Firstly, for the 5U(3) case, we retain the fermions a,b but replace the operators E) by the corresponding expressions in terms of bosonic creation/annihilation operators,

3 [ . J]- 8J i _ t (13) Y" I" - i, Y =Yi, E) = yiYj - 8)NB' NB = 2. yiYi. i=1

For simplicity write YI == Y, Y2 == S, Y3 == t and note that the combinations

2Tl =E1+Ei, 2T2 = i(Ei - E1),

2T4 =E1 +Er, 2Ts = i(Ef -E1),

2T6=E~+E~, 2T7 = i(E~ -E~),

2T3 =Et -E~, 2T8 =-J3E~, (14)

obey the standard 5U(3) commutation rules

[Ta, Tb] = ifabcTc

with totally antisymmetric structure constants (Lichtenberg 1978). As before, v-spin is generated by the fermion bilinears with Casimir operator

v2 = 3NF(2 -NF)/4, (IS)

while V-spin is generated in analogous way by sand t bilinears [or by T6, T7 and (T3 + T8/J3)/2 if one prefers] with Casimir invariant

v2 = (NB - yt Y)(NB - yt Y+ 2)/4 (16)

from which we identify

p=nB, q = ny+ns = nB-nt and Y= ny,

in comparison with the Gel'fand-Tseytlin labelling scheme for these totally symmetric representations; V = (nB - ny)/2 from (15). The Hilbert space is just the direct sum of irreducible representations of the type {p,O} discussed previously, with all values p=O,I,2, ..., and supercharges

Q = ay t s + by t t, Q t = at sty + b t tty, (17) Supersymmetric Quantum Mechanics 361

leading to the Hamiltonian

H = Nr(2 +NB -Nr -NF) + NaNs + NbNr +v+tt s+v_st t. (18)

This H is at most quartic in the creation and annihilation operators, although the supercharges were already trilinear. This shows in a simplistic toy model that the ansatz of de Crombrugghe and Rittenberg (1983) is certainly not general enough, because supercharges linear in both the fermionic and the bosonic creation and annihilation operators apparently define only an interesting subclass of quartic 5U5Y Hamiltonians. Another hint on necessary extensions of this subclass has been given by Baake et al. (1985). As mentioned in the Introduction, there is a second natural class of model supersymmetric Hamiltonians with the same general features as those discussed so far; namely the case where canonical bosons are coupled in Q and Q t to fermionic generators in a Lie superalgebra. In the n = 2 case one would have, for instance,

Q = r.a t h + s.a t e, Q t = r t .h t a + s t .e t a, (19)

where the bosonic r,s operators are coupled to the odd generators of 5U(2/1) formed from bilinears in the boson a and the fermions h,e. However, as can be seen by comparison with (17), this n = 2 case has the same structure as in the previous oscillator representation, given that the a,r,s boson bilinears generate 5U(3) while the fermion h,e bilinears generate 5U(2). In simple n > 2 generalisations of (2) the same situation prevails, although for higher n different possible types of realisation of the (super)algebra and different choices of (anti)commuting generators coupled in Q and Q t require each case to be examined separately.

4. Conclusions An algebraic perspective of supersymmetric quantum mechanics has been taken in which the structure of the supercharges has been analysed in terms of couplings between bosonic and fermionic modes. The n = 1 and n = 2 cases have been treated for the compact 5U(2) and 5U(3) algebras for fixed irreducible representations as well as in more general Fock spaces arising from oscillator descriptions. The ansatz adopted in the present work is intermediate between the highly constrained supersymmetric potential models (Rittenberg and Yankielowicz 1985; 5ukumar 1985) and models of dynamical superalgebras (Kostelecky and Campbell 1985). From the algebraic viewpoint it can be observed that the embeddings of supersymmetric quantum mechanics are in fact ubiquitous in appropriate superalgebras, * and it is contended that models with this richness may be rather interesting .

• A generalisation of the present ansatz is the coupling of arbitrary anticommuting generators of a Lie superalgebra to commuting generators of a Lie algebra. The simplest possibility is Q = EO( where oc is an odd root and 20c is not a root, but in this case the boson/fermion coupling is not explicit. See Balantekin (1985) for such instances as OSp(1/2) and OSp(2/2). 362 M. Baake et al.

One case of physical interest is the so-called Jaynes-Cummings model (Chaichian et al. 1990). Up to a factor, the Hamiltonian here is the operator

S = b t J- + bJ+, which can be identified in the obvious way with the sum S = Q +Qt of the two supercharges, each of the form (2) and leading to the non-compact dynamical algebra SUO,l). Since H={Q,Qt}=S2, the analysis of H carries over to S=.JFi. Finally we may point out that the present model bears some relation to the BRST construction (Van Holten 1990). There, however, one couples fermionic ghost fields with every root vector of the bosonic Lie algebra; Q contains additional trilinear ghost terms such that the Hamiltonian is proportional to the Casimir invariant and hence remains constant on any fixed irreducible representation.

Acknowledgments This research was supported by a University of Tasmania Research Grant and by Deutsche Forschungsgemeinschaft.

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Manuscript received 24 December 1990, accepted 21 March 1991